1368:
1687:
4739:
1032:
1404:
1363:{\displaystyle {\begin{pmatrix}1&1&1&1&1&0&0&0&0&0\\1&1&0&0&0&1&1&1&0&0\\1&0&1&0&0&1&0&0&1&1\\0&1&0&1&0&0&1&0&1&1\\0&0&1&0&1&0&1&1&1&0\\0&0&0&1&1&1&0&1&0&1\\\end{pmatrix}}}
4964:
4463:
2059:= 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points. They are similar to finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order
1682:{\displaystyle \left({\begin{matrix}1&1&1&0&0&0&0\\1&0&0&1&1&0&0\\1&0&0&0&0&1&1\\0&1&0&1&0&1&0\\0&1&0&0&1&0&1\\0&0&1&1&0&0&1\\0&0&1&0&1&1&0\end{matrix}}\right)}
4734:{\displaystyle {\begin{pmatrix}1&1&1&1&0&0&0&0\\1&1&0&0&1&1&0&0\\0&0&1&1&1&1&0&0\\1&0&1&0&0&0&1&1\\0&1&0&0&1&0&1&1\\0&0&0&1&0&1&1&1\\\end{pmatrix}}}
4748:
967:
given block) constant. For other designs such as partially balanced incomplete block designs this may however be possible. Many such cases are discussed in. However, it can also be observed trivially for the magic squares or magic rectangles which can be viewed as the partially balanced incomplete block designs.
5409:
Suppose that skin cancer researchers want to test three different sunscreens. They coat two different sunscreens on the upper sides of the hands of a test person. After a UV radiation they record the skin irritation in terms of sunburn. The number of treatments is 3 (sunscreens) and the block size is
3264:
Note that the projective plane of order two is an
Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric
966:
A rather surprising and not very obvious (but very general) combinatorial result for these designs is that if points are denoted by any arbitrarily chosen set of equally or unequally spaced numerics, there is no choice of such a set which can make all block-sums (that is, sum of all points in a
2720:
4959:{\displaystyle {\begin{pmatrix}4&2&2&2&1&1\\2&4&2&1&2&1\\2&2&4&1&1&2\\2&1&1&4&2&2\\1&2&1&2&4&2\\1&1&2&2&2&4\\\end{pmatrix}}}
1377:
0123 0124 0156 0257 0345 0367 0467 1267 1346 1357 1457 2347 2356
1995:
of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane,
2952:
5377:
While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in
2038:
Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.
2832:
5162:
6838:: Databases of combinatorial, statistical, and experimental block designs. Software and other resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.
5309:
5232:
2574:
3689:
5845:. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of
5082:
5861:, Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as
3605:
3987:
1817:
777:
2098:(Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the
7084:
3868:
3823:
3752:
1913:
6704:
6673:
3903:
999:
012 013 024 035 045 125 134 145 234 235.
3778:
696:
5021:
378:
2368:
2339:
6892:
5675:
It is impossible to use a complete design (all treatments in each block) in this example because there are 3 sunscreens to test, but only 2 hands on each person.
2394:
1382:
The unique (7,3,1)-design is symmetric and has 7 blocks with each element repeated 3 times. Using the symbols 0 − 6, the blocks are the following triples:
1373:
One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times. Using the symbols 0 − 7 the blocks are the following 4-tuples:
5330:
PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs. They fall into six types based on a classification of the
296:
design only when the design is also binary. The incidence matrix of a non-binary design lists the number of times each element is repeated in each block.
2399:
This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an
Hadamard matrix of size 4
1398:
to the points and lines of the plane. Its corresponding incidence matrix can also be symmetric, if the labels or blocks are sorted the right way:
2837:
5778:
of order six. The 2-design with the indicated parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.
3314: + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,
6819:
6629:
6606:
6584:
6545:
6414:
6322:
6790:
7099:
7053:
1829:
2749:
532:, so that no block contains all the elements of the set. This is the meaning of "incomplete" in the name of these designs.) In a table:
7043:
7013:
6885:
6800:
6773:
6563:
6486:
6464:
6392:
6334:
6291:
5738:
3274:
250:. The designs discussed in this article are all uniform. Block designs that are not necessarily uniform have also been studied; for
7089:
6359:; Shimamoto, T. (1952), "Classification and analysis of partially balanced incomplete block designs with two associate classes",
7120:
6742:
6648:
6259:
2157:
7156:
5668:) contains the treatments 1 and 2 simultaneously and the same applies to the pairs of treatments (1,3) and (2,3). Therefore,
5088:
2273:(that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size
7146:
7063:
6878:
6845:
3346:
7079:
7094:
1697:
The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a
398:
3411:
3302:
2715:{\displaystyle \lambda _{i}=\lambda \left.{\binom {v-i}{t-i}}\right/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}
1832:
gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
1720:, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in
16:
This article is about block designs with fixed block size (uniform). For block designs with variable block sizes, see
6671:
Khattree, Ravindra (2019). "A note on the nonexistence of the constant block-sum balanced incomplete block designs".
5238:
5168:
6313:
3300:
It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An
2202:
2288:
in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix
65:, chosen such that frequency of the elements satisfies certain conditions making the collection of blocks exhibit
7151:
5398:
3617:
6194:
1386:
013 026 045 124 156 235 346.
6995:
5414:
1701:. Symmetric designs have the smallest number of blocks among all the 2-designs with the same number of points.
329:
5027:
6423:
Fisher, R.A. (1940), "An examination of the different possible solutions of a problem in incomplete blocks",
7028:
3547:
2185:
2168:
29:
3936:
2105:
The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with
1763:
723:
7141:
7038:
5545:
blocks, that is, 3 test people in order to obtain a balanced incomplete block design. Labeling the blocks
1748:-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (
948:
5527:
for the block design which are then inserted into the R-function. Subsequently, the remaining parameters
2419:), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a
7115:
6983:
6737:
5394:
5367:
3428:
is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,
2461:
285:
113:
21:
2453:
Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.
36:
7048:
7018:
6958:
6936:
6624:, Carus Mathematical Monographs, vol. 14, Mathematical Association of America, pp. 96–130,
6216:
5371:
3121:) design. Note that derived designs with respect to different points may not be isomorphic. A design
2181:
255:
17:
3828:
3783:
3712:
7058:
7023:
7003:
6901:
6782:
6594:
6572:
317:
70:
54:
7033:
6953:
6721:
6690:
6524:
6425:
6232:
6206:
5684:
3480:
3465:
2148:
1868:
221:
90:
78:
3873:
2127:
The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the
2180:
There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). One is the
6853:
6815:
6796:
6769:
6625:
6602:
6580:
6559:
6541:
6482:
6460:
6410:
6388:
6318:
6301:
6287:
6171:
Not a mathematical classification since one of the types is a catch-all "and everything else".
5757:
3608:
3469:
2113:= 3. Geometrically, the points are the vertices of a tetrahedron and the blocks are its faces.
266:
261:
Block designs may or may not have repeated blocks. Designs without repeated blocks are called
6617:
6963:
6915:
6751:
6733:
6713:
6682:
6657:
6508:
6442:
6434:
6368:
6343:
6268:
6224:
5939:
5747:
5588:
5386:
5379:
3757:
3193:
2457:
2293:
1849:
1844:
1004:
666:
388:
82:
6702:
Khattree, Ravindra (2022). "On construction of equireplicated constant block-sum designs".
6520:
4994:
351:
6516:
3499:
2344:
2315:
2214:
861:. These conditions are not sufficient as, for example, a (43,7,1)-design does not exist.
395:
392:
112:
which has been the most intensely studied type historically due to its application in the
74:
6841:
3290:
2373:
288:). There, a design in which each element occurs the same total number of times is called
6220:
6946:
6786:
6438:
6403:
6381:
5689:
3396:
3031:
2120:: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the
884:
of a 2-design is obtained by replacing each block with its complement in the point set
58:
6756:
7135:
6725:
6694:
6662:
6639:
6528:
6305:
6273:
6254:
6228:
5853:, Springer, 1968), in analogy with the most common example, projective planes, while
5366:
The mathematical subject of block designs originated in the statistical framework of
2140:
2136:
1015:
952:
384:
43:
2197:
Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).
2194:
There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).
6475:
6372:
6236:
5775:
86:
6717:
6686:
2191:
There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).
6856:
5397:. The rows of their incidence matrices are also used as the symbols in a form of
2568:
and the four numbers themselves cannot be chosen arbitrarily. The equations are
6740:(1970), "Non-isomorphic solutions of some balanced incomplete block designs I",
6495:
Kaski, Petteri; Östergård, Patric (2008). "There Are
Exactly Five Biplanes with
6197:(Jul 2012). "Expurgated PPM Using Symmetric Balanced Incomplete Block Designs".
4083:(3) be the following association scheme with three associate classes on the set
1725:
822:) even without assuming it explicitly, thus proving that the condition that any
391:
of a regular uniform block design. Also, each configuration has a corresponding
46:
6332:(1949), "A Note on Fisher's Inequality for Balanced Incomplete Block Designs",
5422:
2143:
of order 11, which is constructed using the field with 11 elements, and is the
7008:
6973:
6920:
6356:
6348:
6329:
5752:
5733:
5390:
2117:
2020:
1395:
1391:
402:
284:
block designs, in which blocks may contain multiple copies of an element (see
277:
25:
5761:
6861:
2012: + 1 is the number of lines with which a given point is incident.
5370:. These designs were especially useful in applications of the technique of
238:
A block design in which all the blocks have the same size (usually denoted
2947:{\displaystyle r=\lambda _{1}=\lambda {v-1 \choose t-1}/{k-1 \choose t-1}}
321:
270:
148:
is unspecified, it can usually be assumed to be 2, which means that each
66:
5774:
Proved by Tarry in 1900 who showed that there was no pair of orthogonal
3436:
is a prime power and there is a unique egglike inversive plane of order
6870:
1940: − 1 and, from the displayed equation above, we obtain
660:
are possible. The two basic equations connecting these parameters are
524:
of the design. (To avoid degenerate examples, it is also assumed that
6512:
6447:
2027:= 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has
995:= 5). Using the symbols 0 − 5, the blocks are the following triples:
152:
of elements is found in the same number of blocks and the design is
69:(balance). Block designs have applications in many areas, including
2156:
Algebraically this corresponds to the exceptional embedding of the
6211:
2827:{\displaystyle b=\lambda _{0}=\lambda {v \choose t}/{k \choose t}}
2099:
6577:
Constructions and
Combinatorial Problems in Design of Experiments
3246: = (λ + 2)(λ + 4λ + 2),
1860:> 1. For these designs the symmetric design equation becomes:
192:=1). When the balancing requirement fails, a design may still be
5374:. This remains a significant area for the use of block designs.
3000:. (Note that the "lambda value" changes as above and depends on
2201:
Biplanes of orders 5, 6, 8 and 10 do not exist, as shown by the
6874:
4071:) determines an association scheme but the converse is false.
3208: + 1, 1) designs) are those of orders 2 and 4.
1835:
The following are important examples of symmetric 2-designs:
3233:,λ) design, is extendable, then one of the following holds:
2415:
is a BIBD whose blocks can be partitioned into sets (called
810:
is a block that contains them both. This equation for every
332:. Such a design is uniform and regular: each block contains
231:, meaning that the collection of blocks is not all possible
5734:"On balanced incomplete-block designs with repeated blocks"
3410:
is an odd power of 2, another type of ovoid is known – the
2635:
2595:
2370:
points/blocks. Each pair of points is contained in exactly
254:=2 they are known in the literature under the general name
6835:
2019:= 2 we get a projective plane of order 2, also called the
160:=1, each element occurs in the same number of blocks (the
140:-subsets of the original set occur in equally many (i.e.,
5897:
5895:
5389:
of block designs provide a natural source of interesting
4430: = 1. Also, for the association scheme we have
3440:. (But it is unknown if non-egglike ones exist.) (b) if
2000:
is the number of points on each line and it is equal to
6599:
Block
Designs: Analysis, Combinatorics and Applications
5157:{\displaystyle \sum _{i=1}^{m}n_{i}\lambda _{i}=r(k-1)}
35:"BIBD" redirects here. For the airport in Iceland, see
5732:
P. Dobcsányi, D.A. Preece. L.H. Soicher (2007-10-01).
4757:
4472:
4032:
blocks, such that there is an association scheme with
1828:, so the number of points is far from arbitrary. The
1413:
1041:
944:. A 2-design and its complement have the same order.
409:
Pairwise balanced uniform designs (2-designs or BIBDs)
6409:(2nd ed.), Boca Raton: Chapman & Hall/ CRC,
5241:
5171:
5091:
5030:
4997:
4751:
4466:
3939:
3876:
3831:
3786:
3760:
3715:
3620:
3550:
2840:
2752:
2577:
2376:
2347:
2318:
1967:
As a projective plane is a symmetric design, we have
1960: + 1 points in a projective plane of order
1871:
1766:
1407:
1035:
726:
669:
588:
number of blocks containing any 2 (or more generally
354:
5941:
From
Biplanes to the Klein quartic and the Buckyball
5538:
Using the basic relations we calculate that we need
3273:
A design with the parameters of the extension of an
2269:
identity matrix. An
Hadamard matrix can be put into
2031: + 1 = 3 points and each point belongs to
380:, which is the total number of element occurrences.
7108:
7072:
6994:
6929:
6908:
6255:"On collineation groups of symmetric block designs"
3452:is egglike (but there may be some unknown ovoids).
3334:
are the blocks of an inversive plane of order
632:)-design. The parameters are not all independent;
280:, the concept of a block design may be extended to
6638:Salwach, Chester J.; Mezzaroba, Joseph A. (1978).
6474:
6402:
6380:
6027:
5938:Martin, Pablo; Singerman, David (April 17, 2008),
5303:
5226:
5156:
5076:
5015:
4958:
4733:
3981:
3897:
3862:
3817:
3772:
3746:
3683:
3599:
3326: + 1 points. The plane sections of size
2946:
2826:
2714:
2388:
2362:
2333:
1907:
1811:
1681:
1362:
771:
690:
372:
6766:Combinatorial Designs: Constructions and Analysis
6705:Communications in Statistics - Theory and Methods
6674:Communications in Statistics - Theory and Methods
6401:Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),
3448:is a power of 2 and any inversive plane of order
3338:. Any inversive plane arising this way is called
2938:
2909:
2895:
2866:
2818:
2805:
2791:
2778:
2671:
2642:
2629:
2600:
2560:,λ)-design. Again, these four numbers determine
2456:Archetypical resolvable 2-designs are the finite
441:, standing for balanced incomplete block design)
5957:
5559:, to avoid confusion, we have the block design,
6361:Journal of the American Statistical Association
6087:
6063:
5304:{\displaystyle n_{i}p_{jh}^{i}=n_{j}p_{ih}^{j}}
5227:{\displaystyle \sum _{u=0}^{m}p_{ju}^{h}=n_{j}}
2147:associated to the size 12 Hadamard matrix; see
6459:(2nd ed.), New York: Wiley-Interscience,
6147:
6099:
5968:
5825:
5707:
5314:A PBIBD(1) is a BIBD and a PBIBD(2) in which λ
3285: + 1, 1) design, is called a finite
849:must be integers, which imposes conditions on
184:-values), so for example a pairwise balanced (
6886:
6848:'s page of web based design theory resources.
5413:A corresponding BIBD can be generated by the
5335:
3211:Every Hadamard 2-design is extendable (to an
2341:blocks/points; each contains/is contained in
2116:The order 2 biplane is the complement of the
1755:The parameters of a symmetric design satisfy
1394:, with the elements and blocks of the design
8:
6159:
6135:
6123:
6111:
3678:
3634:
3594:
3564:
3250: = λ + 3λ + 1,
3007:A consequence of this theorem is that every
991:= 10) and each element is repeated 5 times (
888:. It is also a 2-design and has parameters
265:, in which case the "family" of blocks is a
235:-subsets, thus ruling out a trivial design.
227:Designs are usually said (or assumed) to be
6075:
6051:
6015:
6003:
5901:
4458: = 2. The incidence matrix M is
3684:{\displaystyle R^{*}:=\{(x,y)|(y,x)\in R\}}
3424:be a positive integer, at least 2. (a) If
2548:of the design. The design may be called a
838:can be computed from the other parameters.
6893:
6879:
6871:
6180:
5979:
4006:partially balanced incomplete block design
3342:. All known inversive planes are egglike.
2520:appears in exactly λ blocks. The numbers
701:obtained by counting the number of pairs (
568:number of blocks containing a given point
336:elements and each element is contained in
108:), specifically (and also synonymously) a
6755:
6661:
6481:, Cambridge: Cambridge University Press,
6446:
6347:
6286:, Cambridge: Cambridge University Press,
6272:
6210:
5925:
5751:
5425:and is specified in the following table:
5295:
5287:
5277:
5264:
5256:
5246:
5240:
5218:
5205:
5197:
5187:
5176:
5170:
5127:
5117:
5107:
5096:
5090:
5056:
5046:
5035:
5029:
4996:
4752:
4750:
4467:
4465:
3973:
3965:
3952:
3944:
3938:
3889:
3881:
3875:
3854:
3830:
3809:
3785:
3759:
3738:
3714:
3652:
3625:
3619:
3555:
3549:
3349:, the set of zeros of the quadratic form
3105:which contain p with p removed. It is a (
2937:
2908:
2906:
2901:
2894:
2865:
2863:
2851:
2839:
2817:
2804:
2802:
2797:
2790:
2777:
2775:
2763:
2751:
2732:is the number of blocks that contain any
2677:
2670:
2641:
2639:
2628:
2599:
2597:
2582:
2576:
2375:
2346:
2317:
2124:of the (3-point) lines in the Fano plane.
1870:
1765:
1412:
1406:
1036:
1034:
725:
668:
387:with constant row and column sums is the
353:
6538:Symmetric Designs: An Algebraic Approach
6379:Cameron, P. J.; van Lint, J. H. (1991),
5809:
5798:
5787:
5593:
5509:The investigator chooses the parameters
5427:
5077:{\displaystyle \sum _{i=1}^{m}n_{i}=v-1}
4113:
3265:2-design with the parameters of case 3.
2464:is a resolution of a 2-(15,3,1) design.
2094:The 18 known examples are listed below.
536:
469:blocks, and any pair of distinct points
304:The simplest type of "balanced" design (
300:Regular uniform designs (configurations)
176:is also balanced in all lower values of
96:Without further specifications the term
6039:
5991:
5874:
5719:
5700:
4060:, then they are together in precisely λ
2308: − 1) design called an
2144:
204:classes, each with its own (different)
6383:Designs, Graphs, Codes and their Links
5859:Designs, Graphs, Codes and their Links
5654:Each treatment occurs in 2 blocks, so
3600:{\displaystyle R_{0}=\{(x,x):x\in X\}}
6810:van Lint, J.H.; Wilson, R.M. (1992).
5886:
5591:is specified in the following table:
4402:The parameters of this PBIBD(3) are:
4001:. Most authors assume this property.
3982:{\displaystyle p_{ij}^{k}=p_{ji}^{k}}
3101:} and as block set all the blocks of
1812:{\displaystyle \lambda (v-1)=k(k-1).}
772:{\displaystyle \lambda (v-1)=r(k-1),}
497:blocks is redundant, as shown below.
485:blocks. Here, the condition that any
7:
6792:Combinatorics of Experimental Design
6554:Lindner, C.C.; Rodger, C.A. (1997),
5913:
5821:
5819:
5817:
3917:but not on the particular choice of
3041:by itself usually means a 2-design.
2035: + 1 = 3 lines.
1824:This imposes strong restrictions on
57:consisting of a set together with a
6597:; Padgett, L.V. (11 October 2005).
5837:They have also been referred to as
4028:and with each element appearing in
3456:Partially balanced designs (PBIBDs)
2169:projective linear group: action on
1390:This design is associated with the
782:obtained from counting for a fixed
340:blocks. The number of set elements
116:. Its generalization is known as a
6473:Hughes, D.R.; Piper, E.C. (1985),
6439:10.1111/j.1469-1809.1940.tb02237.x
4363:The blocks of a PBIBD(3) based on
3514:. A pair of elements in relation R
2913:
2870:
2809:
2782:
2646:
2604:
2284:Given an Hadamard matrix of size 4
14:
6405:Handbook of Combinatorial Designs
6335:Annals of Mathematical Statistics
5739:European Journal of Combinatorics
4743:and the concurrence matrix MM is
3260: = 39, λ = 3.
6501:Journal of Combinatorial Designs
6282:Assmus, E.F.; Key, J.D. (1992),
6229:10.1109/LCOMM.2012.042512.120457
6028:Beth, Jungnickel & Lenz 1986
4016:)) is a block design based on a
3177:) design has an extension, then
3045:Derived and extendable t-designs
2083: + 1)/2 points (since
1987: + 1 also. The number
102:balanced incomplete block design
6743:Journal of Combinatorial Theory
6649:Journal of Combinatorial Theory
6260:Journal of Combinatorial Theory
2233:whose entries are ±1 such that
2184:. These three designs are also
2158:projective special linear group
1752:) is a symmetric block design.
951:, named after the statistician
868:of a 2-design is defined to be
168:) and the design is said to be
7044:Cremona–Richmond configuration
6814:. Cambridge University Press.
6540:, Cambridge University Press,
6387:, Cambridge University Press,
6373:10.1080/01621459.1952.10501161
5535:are determined automatically.
5151:
5139:
4968:from which we can recover the
3863:{\displaystyle (z,y)\in R_{j}}
3844:
3832:
3818:{\displaystyle (x,z)\in R_{i}}
3799:
3787:
3747:{\displaystyle (x,y)\in R_{k}}
3728:
3716:
3669:
3657:
3653:
3649:
3637:
3579:
3567:
3345:An example of an ovoid is the
2071: + 2 points; it has
1899:
1887:
1803:
1791:
1782:
1770:
763:
751:
742:
730:
717:is a point in that block, and
648:, and not all combinations of
545:points, number of elements of
1:
6757:10.1016/S0021-9800(70)80024-2
6718:10.1080/03610926.2020.1814816
6687:10.1080/03610926.2018.1508715
2055:is a symmetric 2-design with
1852:are symmetric 2-designs with
200:-subsets can be divided into
7121:Kirkman's schoolgirl problem
7054:Grünbaum–Rigby configuration
6764:Stinson, Douglas R. (2003),
6663:10.1016/0097-3165(78)90002-X
6616:Ryser, Herbert John (1963),
6274:10.1016/0097-3165(71)90054-9
6253:Aschbacher, Michael (1971).
5958:Salwach & Mezzaroba 1978
5372:analysis of variance (ANOVA)
3541:th associates. Furthermore:
2460:. A solution of the famous
2075: = 1 + (
2004: + 1. Similarly,
1693:Symmetric 2-designs (SBIBDs)
818:is constant (independent of
578:number of points in a block
188:=2) design is also regular (
172:. Any design balanced up to
7014:Möbius–Kantor configuration
6455:Hall, Marshall Jr. (1986),
6199:IEEE Communications Letters
6088:Cameron & van Lint 1991
6064:Cameron & van Lint 1991
5336:Bose & Shimamoto (1952)
2736:-element set of points and
2524:(the number of elements of
2476:Given any positive integer
1930:order of a projective plane
1908:{\displaystyle v-1=k(k-1).}
1728:provides the converse. If
975:The unique (6,3,2)-design (
504:(the number of elements of
7173:
7100:Bruck–Ryser–Chowla theorem
6618:"8. Combinatorial Designs"
6314:Cambridge University Press
6150:, pg. 562, Remark 42.3 (4)
6148:Colbourn & Dinitz 2007
6100:Colbourn & Dinitz 2007
6066:, pg. 11, Proposition 1.34
5969:Kaski & Östergård 2008
5826:Colbourn & Dinitz 2007
5708:Colbourn & Dinitz 2007
5326:Two associate class PBIBDs
4984:The parameters of a PBIBD(
3898:{\displaystyle p_{ij}^{k}}
3395:where f is an irreducible
2468:General balanced designs (
2434:,λ) resolvable design has
2203:Bruck-Ryser-Chowla theorem
2139:; it is associated to the
1842:
1830:Bruck–Ryser–Chowla theorem
401:known as its incidence or
34:
15:
7090:Szemerédi–Trotter theorem
6812:A Course in Combinatorics
6622:Combinatorial Mathematics
6558:, Boca Raton: CRC Press,
5753:10.1016/j.ejc.2006.08.007
5399:pulse-position modulation
4012:associate classes (PBIBD(
3929:An association scheme is
3399:in two variables over GF(
2281:must be a multiple of 4.
2063:is one whose blocks have
344:and the number of blocks
256:pairwise balanced designs
7080:Sylvester–Gallai theorem
6640:"The four biplanes with
6160:Street & Street 1987
6136:Street & Street 1987
6124:Street & Street 1987
6112:Street & Street 1987
6018:, pg. 158, Corollary 5.5
3240:is an Hadamard 2-design,
3155:if it has an extension.
3137:has a point p such that
3015:≥ 2 is also a 2-design.
2532:(the number of blocks),
2500:, such that every point
1850:Finite projective planes
1022:and constant column sum
834:blocks is redundant and
806:are distinct points and
600:The design is called a (
512:(the number of blocks),
330:Configuration (geometry)
220:, whose classes form an
7085:De Bruijn–Erdős theorem
7029:Desargues configuration
6842:Design Theory Resources
6349:10.1214/aoms/1177729958
6284:Designs and Their Codes
6102:, pg. 114, Remarks 6.35
6076:Hughes & Piper 1985
6052:Hughes & Piper 1985
6042:, pg.203, Corollary 9.6
6016:Hughes & Piper 1985
6004:Hughes & Piper 1985
5902:Hughes & Piper 1985
5849:is due to P.Dembowski (
5405:Statistical application
3181: + 1 divides
3109: − 1)-(
3030:,1)-design is called a
2438:parallel classes, then
2300: − 1, 2
2277: > 2 then
947:A fundamental theorem,
180:(though with different
128:A design is said to be
30:randomized block design
6738:Bhat-Nayak, Vasanti N.
6536:Lander, E. S. (1983),
6090:, pg. 11, Theorem 1.35
6078:, pg. 132, Theorem 4.5
6006:, pg. 156, Theorem 5.4
5857:is due to P. Cameron (
5410:2 (hands per person).
5395:error correcting codes
5354:partial geometry type;
5305:
5228:
5192:
5158:
5112:
5078:
5051:
5017:
4960:
4735:
4087:= {1,2,3,4,5,6}. The (
3983:
3899:
3864:
3819:
3774:
3773:{\displaystyle z\in X}
3748:
3685:
3601:
3117: − 1,
3113: − 1,
2972:,λ)-design is also an
2948:
2828:
2716:
2450: − 1.
2390:
2364:
2335:
2304: − 1,
1909:
1813:
1704:In a symmetric design
1683:
1364:
1018:with constant row sum
1003:and the corresponding
773:
692:
691:{\displaystyle bk=vr,}
374:
310:tactical configuration
212:=2 these are known as
7157:Design of experiments
7116:Design of experiments
5994:, pg. 74, Theorem 4.5
5926:Assmus & Key 1992
5368:design of experiments
5306:
5229:
5172:
5159:
5092:
5079:
5031:
5018:
5016:{\displaystyle vr=bk}
4961:
4736:
4418: = 4 and λ
3984:
3900:
3865:
3820:
3775:
3749:
3686:
3602:
2949:
2829:
2717:
2462:15 schoolgirl problem
2391:
2365:
2336:
1910:
1814:
1724:points. A theorem of
1684:
1365:
774:
693:
375:
373:{\displaystyle bk=vr}
324:is known simply as a
286:blocking (statistics)
114:design of experiments
7147:Combinatorial design
7049:Kummer configuration
7019:Pappus configuration
6902:Incidence structures
6783:Street, Anne Penfold
6601:. World Scientific.
6595:Raghavarao, Damaraju
6573:Raghavarao, Damaraju
6457:Combinatorial Theory
6195:Brandt-Pearce, Maïté
6030:, pg. 40 Example 5.8
5877:, pg.23, Theorem 2.2
5239:
5169:
5089:
5028:
4995:
4749:
4464:
4444: = 1 and
4426: = 2 and λ
4024:blocks each of size
3937:
3874:
3829:
3784:
3758:
3713:
3618:
3548:
3192:The only extendable
3097: − {
2838:
2750:
2575:
2492:-element subsets of
2407:Resolvable 2-designs
2374:
2363:{\displaystyle 2a-1}
2345:
2334:{\displaystyle 4a-1}
2316:
2248:is the transpose of
2182:Kummer configuration
2149:Paley construction I
1869:
1764:
1405:
1033:
987:= 2) has 10 blocks (
724:
667:
449:-element subsets of
417:(of elements called
352:
316:. The corresponding
100:usually refers to a
22:experimental designs
18:Combinatorial design
7059:Klein configuration
7039:Schläfli double six
7024:Hesse configuration
7004:Complete quadrangle
6221:2012arXiv1203.5378N
5423:R-package agricolae
5300:
5269:
5210:
4052:th associates, 1 ≤
4040:where, if elements
4036:classes defined on
3978:
3957:
3894:
2992:with 1 ≤
2508:appears in exactly
2413:resolvable 2-design
2389:{\displaystyle a-1}
2296:of a symmetric 2-(4
949:Fisher's inequality
876: −
413:Given a finite set
318:incidence structure
71:experimental design
55:incidence structure
7034:Reye configuration
6854:Weisstein, Eric W.
6787:Street, Deborah J.
6426:Annals of Eugenics
6302:Jungnickel, Dieter
6193:Noshad, Mohammad;
6138:, pg. 240, Lemma 4
5839:projective designs
5685:Incidence geometry
5348:Latin square type;
5301:
5283:
5252:
5224:
5193:
5154:
5074:
5013:
4956:
4950:
4731:
4725:
3979:
3961:
3940:
3895:
3877:
3860:
3815:
3770:
3744:
3681:
3607:and is called the
3597:
3526:. Each element of
3466:association scheme
3330: + 1 of
3256: = 495,
2944:
2824:
2712:
2512:blocks, and every
2386:
2360:
2331:
2209:Hadamard 2-designs
1905:
1809:
1736:-element set, and
1679:
1673:
1360:
1354:
769:
688:
640:, and λ determine
592:) distinct points
508:, called points),
445:to be a family of
370:
308:=1) is known as a
222:association scheme
194:partially balanced
162:replication number
91:algebraic geometry
79:physical chemistry
37:Bíldudalur Airport
7129:
7128:
6821:978-0-521-41057-1
6795:. Oxford U. P. .
6681:(20): 5165–5168.
6631:978-1-61444-014-7
6608:978-981-4480-23-9
6586:978-0-486-65685-4
6547:978-0-521-28693-0
6513:10.1002/jcd.20145
6416:978-1-58488-506-1
6323:978-0-521-44432-3
6317:. 2nd ed. (1999)
5851:Finite Geometries
5652:
5651:
5507:
5506:
5393:that are used as
4414: = 3,
4410: = 8,
4406: = 6,
4400:
4399:
4361:
4360:
4107:are in relation R
3609:Identity relation
3412:Suzuki–Tits ovoid
3318:) meets an ovoid
3225:, a symmetric 2-(
3213:Hadamard 3-design
3194:projective planes
3189: + 1).
3144:is isomorphic to
2988:)-design for any
2936:
2893:
2816:
2789:
2680:
2669:
2627:
2310:Hadamard 2-design
2271:standardized form
2145:Hadamard 2-design
1991:is the number of
1952: + 1 =
1928:we can write the
1839:Projective planes
1712:holds as well as
963:in any 2-design.
814:also proves that
596:
595:
558:number of blocks
433:≥ 1, we define a
154:pairwise balanced
59:family of subsets
7164:
7152:Families of sets
6964:Projective plane
6916:Incidence matrix
6895:
6888:
6881:
6872:
6867:
6866:
6836:DesignTheory.Org
6825:
6806:
6778:
6760:
6759:
6734:Shrikhande, S.S.
6729:
6712:(2): 4434–4450.
6698:
6667:
6665:
6634:
6612:
6590:
6568:
6550:
6532:
6491:
6480:
6469:
6451:
6450:
6419:
6408:
6397:
6386:
6375:
6367:(258): 151–184,
6357:Bose, R. C.
6352:
6351:
6316:
6296:
6278:
6276:
6241:
6240:
6214:
6190:
6184:
6178:
6172:
6169:
6163:
6157:
6151:
6145:
6139:
6133:
6127:
6121:
6115:
6109:
6103:
6097:
6091:
6085:
6079:
6073:
6067:
6061:
6055:
6049:
6043:
6037:
6031:
6025:
6019:
6013:
6007:
6001:
5995:
5989:
5983:
5977:
5971:
5966:
5960:
5955:
5949:
5948:
5946:
5935:
5929:
5923:
5917:
5911:
5905:
5899:
5890:
5884:
5878:
5872:
5866:
5835:
5829:
5823:
5812:
5807:
5801:
5796:
5790:
5785:
5779:
5772:
5766:
5765:
5755:
5746:(7): 1955–1970.
5729:
5723:
5717:
5711:
5705:
5671:
5667:
5664:Just one block (
5660:
5594:
5589:incidence matrix
5587:A corresponding
5582:
5575:
5569:},
5568:
5558:
5554:
5544:
5534:
5530:
5526:
5522:
5515:
5428:
5387:incidence matrix
5380:software testing
5342:group divisible;
5310:
5308:
5307:
5302:
5299:
5294:
5282:
5281:
5268:
5263:
5251:
5250:
5233:
5231:
5230:
5225:
5223:
5222:
5209:
5204:
5191:
5186:
5163:
5161:
5160:
5155:
5132:
5131:
5122:
5121:
5111:
5106:
5083:
5081:
5080:
5075:
5061:
5060:
5050:
5045:
5022:
5020:
5019:
5014:
4965:
4963:
4962:
4957:
4955:
4954:
4740:
4738:
4737:
4732:
4730:
4729:
4396: 456
4382: 456
4370:
4369:
4357:
4352:
4347:
4342:
4337:
4332:
4320:
4315:
4310:
4305:
4300:
4295:
4283:
4278:
4273:
4268:
4263:
4258:
4246:
4241:
4236:
4231:
4226:
4221:
4209:
4204:
4199:
4194:
4189:
4184:
4172:
4167:
4162:
4157:
4152:
4147:
4114:
3988:
3986:
3985:
3980:
3977:
3972:
3956:
3951:
3904:
3902:
3901:
3896:
3893:
3888:
3869:
3867:
3866:
3861:
3859:
3858:
3824:
3822:
3821:
3816:
3814:
3813:
3779:
3777:
3776:
3771:
3754:, the number of
3753:
3751:
3750:
3745:
3743:
3742:
3690:
3688:
3687:
3682:
3656:
3630:
3629:
3606:
3604:
3603:
3598:
3560:
3559:
3500:binary relations
3479:together with a
3347:elliptic quadric
3293:, of order
3281: + 1,
3269:Inversive planes
3204: + 1,
2953:
2951:
2950:
2945:
2943:
2942:
2941:
2935:
2924:
2912:
2905:
2900:
2899:
2898:
2892:
2881:
2869:
2856:
2855:
2833:
2831:
2830:
2825:
2823:
2822:
2821:
2808:
2801:
2796:
2795:
2794:
2781:
2768:
2767:
2721:
2719:
2718:
2713:
2681:
2678:
2676:
2675:
2674:
2668:
2657:
2645:
2638:
2634:
2633:
2632:
2626:
2615:
2603:
2587:
2586:
2516:-element subset
2417:parallel classes
2395:
2393:
2392:
2387:
2369:
2367:
2366:
2361:
2340:
2338:
2337:
2332:
2294:incidence matrix
2133:
2132:
2079: + 2)(
2053:biplane geometry
1914:
1912:
1911:
1906:
1845:Projective plane
1818:
1816:
1815:
1810:
1744:-element set of
1699:symmetric design
1688:
1686:
1685:
1680:
1678:
1674:
1369:
1367:
1366:
1361:
1359:
1358:
1005:incidence matrix
830:is contained in
778:
776:
775:
770:
697:
695:
694:
689:
537:
520:, and λ are the
493:is contained in
481:is contained in
465:is contained in
457:, such that any
389:incidence matrix
379:
377:
376:
371:
292:which implies a
83:software testing
7172:
7171:
7167:
7166:
7165:
7163:
7162:
7161:
7132:
7131:
7130:
7125:
7104:
7068:
6990:
6925:
6921:Incidence graph
6904:
6899:
6857:"Block Designs"
6852:
6851:
6832:
6822:
6809:
6803:
6781:
6776:
6763:
6732:
6701:
6670:
6637:
6632:
6615:
6609:
6593:
6587:
6571:
6566:
6553:
6548:
6535:
6494:
6489:
6472:
6467:
6454:
6422:
6417:
6400:
6395:
6378:
6355:
6328:
6299:
6294:
6281:
6252:
6249:
6244:
6192:
6191:
6187:
6181:Raghavarao 1988
6179:
6175:
6170:
6166:
6158:
6154:
6146:
6142:
6134:
6130:
6122:
6118:
6110:
6106:
6098:
6094:
6086:
6082:
6074:
6070:
6062:
6058:
6050:
6046:
6038:
6034:
6026:
6022:
6014:
6010:
6002:
5998:
5990:
5986:
5980:Aschbacher 1971
5978:
5974:
5967:
5963:
5956:
5952:
5944:
5937:
5936:
5932:
5924:
5920:
5912:
5908:
5900:
5893:
5885:
5881:
5873:
5869:
5836:
5832:
5824:
5815:
5808:
5804:
5797:
5793:
5786:
5782:
5773:
5769:
5731:
5730:
5726:
5718:
5714:
5706:
5702:
5698:
5681:
5669:
5665:
5655:
5577:
5570:
5563:
5556:
5546:
5539:
5532:
5528:
5524:
5517:
5510:
5407:
5364:
5328:
5321:
5318: = λ
5317:
5273:
5242:
5237:
5236:
5214:
5167:
5166:
5123:
5113:
5087:
5086:
5052:
5026:
5025:
4993:
4992:
4982:
4966:
4949:
4948:
4943:
4938:
4933:
4928:
4923:
4917:
4916:
4911:
4906:
4901:
4896:
4891:
4885:
4884:
4879:
4874:
4869:
4864:
4859:
4853:
4852:
4847:
4842:
4837:
4832:
4827:
4821:
4820:
4815:
4810:
4805:
4800:
4795:
4789:
4788:
4783:
4778:
4773:
4768:
4763:
4753:
4747:
4746:
4741:
4724:
4723:
4718:
4713:
4708:
4703:
4698:
4693:
4688:
4682:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4640:
4639:
4634:
4629:
4624:
4619:
4614:
4609:
4604:
4598:
4597:
4592:
4587:
4582:
4577:
4572:
4567:
4562:
4556:
4555:
4550:
4545:
4540:
4535:
4530:
4525:
4520:
4514:
4513:
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4468:
4462:
4461:
4457:
4450:
4443:
4436:
4429:
4425:
4421:
4393: 236
4390: 136
4387: 125
4379: 235
4376: 134
4373: 124
4355:
4350:
4345:
4340:
4335:
4330:
4318:
4313:
4308:
4303:
4298:
4293:
4281:
4276:
4271:
4266:
4261:
4256:
4244:
4239:
4234:
4229:
4224:
4219:
4207:
4202:
4197:
4192:
4187:
4182:
4170:
4165:
4160:
4155:
4150:
4145:
4110:
4077:
4063:
3935:
3934:
3872:
3871:
3850:
3827:
3826:
3805:
3782:
3781:
3756:
3755:
3734:
3711:
3710:
3621:
3616:
3615:
3551:
3546:
3545:
3536:
3518:are said to be
3517:
3513:
3509:
3505:
3458:
3386:
3379:
3368:
3362:
3322:in either 1 or
3287:inversive plane
3271:
3143:
3092:
3047:
2987:
2925:
2914:
2907:
2882:
2871:
2864:
2847:
2836:
2835:
2803:
2776:
2759:
2748:
2747:
2741:
2730:
2679: for
2658:
2647:
2640:
2616:
2605:
2598:
2594:
2578:
2573:
2572:
2474:
2423:of the design.
2409:
2372:
2371:
2343:
2342:
2314:
2313:
2312:. It contains
2260:
2243:
2215:Hadamard matrix
2211:
2130:
2129:
2045:
1975:, meaning that
1948: + 1)
1867:
1866:
1847:
1841:
1762:
1761:
1695:
1672:
1671:
1666:
1661:
1656:
1651:
1646:
1641:
1635:
1634:
1629:
1624:
1619:
1614:
1609:
1604:
1598:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1561:
1560:
1555:
1550:
1545:
1540:
1535:
1530:
1524:
1523:
1518:
1513:
1508:
1503:
1498:
1493:
1487:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1450:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1408:
1403:
1402:
1353:
1352:
1347:
1342:
1337:
1332:
1327:
1322:
1317:
1312:
1307:
1301:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1260:
1255:
1249:
1248:
1243:
1238:
1233:
1228:
1223:
1218:
1213:
1208:
1203:
1197:
1196:
1191:
1186:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1145:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1093:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1037:
1031:
1030:
973:
722:
721:
713:is a block and
665:
664:
612:)-design or a (
421:) and integers
411:
350:
349:
348:are related by
302:
144:) blocks. When
126:
75:finite geometry
40:
33:
12:
11:
5:
7170:
7168:
7160:
7159:
7154:
7149:
7144:
7134:
7133:
7127:
7126:
7124:
7123:
7118:
7112:
7110:
7106:
7105:
7103:
7102:
7097:
7095:Beck's theorem
7092:
7087:
7082:
7076:
7074:
7070:
7069:
7067:
7066:
7061:
7056:
7051:
7046:
7041:
7036:
7031:
7026:
7021:
7016:
7011:
7006:
7000:
6998:
6996:Configurations
6992:
6991:
6989:
6988:
6987:
6986:
6978:
6977:
6976:
6968:
6967:
6966:
6961:
6951:
6950:
6949:
6947:Steiner system
6944:
6933:
6931:
6927:
6926:
6924:
6923:
6918:
6912:
6910:
6909:Representation
6906:
6905:
6900:
6898:
6897:
6890:
6883:
6875:
6869:
6868:
6849:
6839:
6831:
6830:External links
6828:
6827:
6826:
6820:
6807:
6801:
6779:
6774:
6761:
6750:(2): 174–191,
6730:
6699:
6668:
6656:(2): 141–145.
6635:
6630:
6613:
6607:
6591:
6585:
6569:
6564:
6551:
6546:
6533:
6507:(2): 117–127.
6492:
6487:
6470:
6465:
6452:
6420:
6415:
6398:
6393:
6376:
6353:
6342:(4): 619–620,
6326:
6306:Lenz, Hanfried
6300:Beth, Thomas;
6297:
6292:
6279:
6267:(3): 272–281.
6248:
6245:
6243:
6242:
6205:(7): 968–971.
6185:
6173:
6164:
6152:
6140:
6128:
6116:
6104:
6092:
6080:
6068:
6056:
6044:
6032:
6020:
6008:
5996:
5984:
5972:
5961:
5950:
5930:
5918:
5906:
5891:
5879:
5867:
5843:square designs
5830:
5813:
5802:
5791:
5780:
5767:
5724:
5712:
5699:
5697:
5694:
5693:
5692:
5690:Steiner system
5687:
5680:
5677:
5650:
5649:
5646:
5643:
5640:
5636:
5635:
5632:
5629:
5626:
5622:
5621:
5618:
5615:
5612:
5608:
5607:
5604:
5601:
5598:
5585:
5584:
5505:
5504:
5501:
5498:
5494:
5493:
5490:
5487:
5483:
5482:
5479:
5476:
5472:
5471:
5468:
5465:
5461:
5460:
5457:
5454:
5450:
5449:
5446:
5443:
5439:
5438:
5435:
5432:
5406:
5403:
5363:
5360:
5359:
5358:
5357:miscellaneous.
5355:
5352:
5349:
5346:
5343:
5327:
5324:
5319:
5315:
5312:
5311:
5298:
5293:
5290:
5286:
5280:
5276:
5272:
5267:
5262:
5259:
5255:
5249:
5245:
5234:
5221:
5217:
5213:
5208:
5203:
5200:
5196:
5190:
5185:
5182:
5179:
5175:
5164:
5153:
5150:
5147:
5144:
5141:
5138:
5135:
5130:
5126:
5120:
5116:
5110:
5105:
5102:
5099:
5095:
5084:
5073:
5070:
5067:
5064:
5059:
5055:
5049:
5044:
5041:
5038:
5034:
5023:
5012:
5009:
5006:
5003:
5000:
4981:
4978:
4953:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4918:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4886:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4854:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4822:
4819:
4816:
4814:
4811:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4791:
4790:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4758:
4756:
4745:
4728:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4684:
4683:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4641:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4599:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4557:
4554:
4551:
4549:
4546:
4544:
4541:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4473:
4471:
4460:
4455:
4451: =
4448:
4441:
4437: =
4434:
4427:
4423:
4422: = λ
4419:
4398:
4397:
4394:
4391:
4388:
4384:
4383:
4380:
4377:
4374:
4359:
4358:
4353:
4351: 1
4348:
4346: 1
4343:
4338:
4333:
4331: 3
4328:
4322:
4321:
4319: 1
4316:
4311:
4309: 1
4306:
4301:
4296:
4291:
4285:
4284:
4282: 1
4279:
4277: 1
4274:
4269:
4264:
4259:
4254:
4248:
4247:
4242:
4237:
4232:
4227:
4225: 1
4222:
4220: 1
4217:
4211:
4210:
4205:
4200:
4195:
4193: 1
4190:
4188: 0
4185:
4183: 1
4180:
4174:
4173:
4168:
4163:
4161: 2
4158:
4156: 1
4153:
4151: 1
4148:
4143:
4137:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4108:
4076:
4073:
4061:
3976:
3971:
3968:
3964:
3960:
3955:
3950:
3947:
3943:
3927:
3926:
3892:
3887:
3884:
3880:
3870:is a constant
3857:
3853:
3849:
3846:
3843:
3840:
3837:
3834:
3812:
3808:
3804:
3801:
3798:
3795:
3792:
3789:
3769:
3766:
3763:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3707:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3655:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3628:
3624:
3612:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3558:
3554:
3534:
3515:
3511:
3507:
3503:
3468:consists of a
3457:
3454:
3444:is even, then
3397:quadratic form
3393:
3392:
3391:
3390:
3389:
3388:
3384:
3377:
3366:
3360:
3310:) is a set of
3270:
3267:
3262:
3261:
3251:
3241:
3196:(symmetric 2-(
3141:
3093:has point set
3088:
3083:derived design
3046:
3043:
3032:Steiner system
2985:
2940:
2934:
2931:
2928:
2923:
2920:
2917:
2911:
2904:
2897:
2891:
2888:
2885:
2880:
2877:
2874:
2868:
2862:
2859:
2854:
2850:
2846:
2843:
2820:
2815:
2812:
2807:
2800:
2793:
2788:
2785:
2780:
2774:
2771:
2766:
2762:
2758:
2755:
2739:
2728:
2723:
2722:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2673:
2667:
2664:
2661:
2656:
2653:
2650:
2644:
2637:
2631:
2625:
2622:
2619:
2614:
2611:
2608:
2602:
2596:
2593:
2590:
2585:
2581:
2488:is a class of
2473:
2466:
2408:
2405:
2385:
2382:
2379:
2359:
2356:
2353:
2350:
2330:
2327:
2324:
2321:
2256:
2241:
2237: = m
2210:
2207:
2199:
2198:
2195:
2192:
2189:
2177:
2176:
2153:
2152:
2125:
2114:
2103:
2044:
2041:
1918:
1917:
1916:
1915:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1856:= 1 and order
1843:Main article:
1840:
1837:
1822:
1821:
1820:
1819:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1694:
1691:
1690:
1689:
1677:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1636:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1599:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1562:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1525:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1488:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1451:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1414:
1411:
1388:
1387:
1380:
1379:
1371:
1370:
1357:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1302:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1250:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1198:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1146:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1094:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1042:
1040:
1001:
1000:
972:
969:
940: − 2
841:The resulting
780:
779:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
699:
698:
687:
684:
681:
678:
675:
672:
598:
597:
594:
593:
586:
580:
579:
576:
570:
569:
566:
560:
559:
556:
550:
549:
543:
410:
407:
369:
366:
363:
360:
357:
301:
298:
290:equireplicate,
269:rather than a
125:
122:
13:
10:
9:
6:
4:
3:
2:
7169:
7158:
7155:
7153:
7150:
7148:
7145:
7143:
7142:Combinatorics
7140:
7139:
7137:
7122:
7119:
7117:
7114:
7113:
7111:
7107:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7077:
7075:
7071:
7065:
7062:
7060:
7057:
7055:
7052:
7050:
7047:
7045:
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7025:
7022:
7020:
7017:
7015:
7012:
7010:
7007:
7005:
7002:
7001:
6999:
6997:
6993:
6985:
6982:
6981:
6979:
6975:
6972:
6971:
6970:Graph theory
6969:
6965:
6962:
6960:
6957:
6956:
6955:
6952:
6948:
6945:
6943:
6940:
6939:
6938:
6937:Combinatorics
6935:
6934:
6932:
6928:
6922:
6919:
6917:
6914:
6913:
6911:
6907:
6903:
6896:
6891:
6889:
6884:
6882:
6877:
6876:
6873:
6864:
6863:
6858:
6855:
6850:
6847:
6846:Peter Cameron
6843:
6840:
6837:
6834:
6833:
6829:
6823:
6817:
6813:
6808:
6804:
6802:0-19-853256-3
6798:
6794:
6793:
6788:
6784:
6780:
6777:
6775:0-387-95487-2
6771:
6767:
6762:
6758:
6753:
6749:
6745:
6744:
6739:
6735:
6731:
6727:
6723:
6719:
6715:
6711:
6707:
6706:
6700:
6696:
6692:
6688:
6684:
6680:
6676:
6675:
6669:
6664:
6659:
6655:
6651:
6650:
6645:
6643:
6636:
6633:
6627:
6623:
6619:
6614:
6610:
6604:
6600:
6596:
6592:
6588:
6582:
6578:
6574:
6570:
6567:
6565:0-8493-3986-3
6561:
6557:
6556:Design Theory
6552:
6549:
6543:
6539:
6534:
6530:
6526:
6522:
6518:
6514:
6510:
6506:
6502:
6498:
6493:
6490:
6488:0-521-25754-9
6484:
6479:
6478:
6477:Design theory
6471:
6468:
6466:0-471-09138-3
6462:
6458:
6453:
6449:
6444:
6440:
6436:
6432:
6428:
6427:
6421:
6418:
6412:
6407:
6406:
6399:
6396:
6394:0-521-42385-6
6390:
6385:
6384:
6377:
6374:
6370:
6366:
6362:
6358:
6354:
6350:
6345:
6341:
6337:
6336:
6331:
6327:
6324:
6320:
6315:
6311:
6310:Design Theory
6307:
6303:
6298:
6295:
6293:0-521-41361-3
6289:
6285:
6280:
6275:
6270:
6266:
6262:
6261:
6256:
6251:
6250:
6246:
6238:
6234:
6230:
6226:
6222:
6218:
6213:
6208:
6204:
6200:
6196:
6189:
6186:
6182:
6177:
6174:
6168:
6165:
6161:
6156:
6153:
6149:
6144:
6141:
6137:
6132:
6129:
6125:
6120:
6117:
6113:
6108:
6105:
6101:
6096:
6093:
6089:
6084:
6081:
6077:
6072:
6069:
6065:
6060:
6057:
6053:
6048:
6045:
6041:
6036:
6033:
6029:
6024:
6021:
6017:
6012:
6009:
6005:
6000:
5997:
5993:
5988:
5985:
5982:, pp. 279–281
5981:
5976:
5973:
5970:
5965:
5962:
5959:
5954:
5951:
5943:
5942:
5934:
5931:
5927:
5922:
5919:
5915:
5910:
5907:
5903:
5898:
5896:
5892:
5889:, pp. 102–104
5888:
5883:
5880:
5876:
5871:
5868:
5864:
5860:
5856:
5852:
5848:
5844:
5840:
5834:
5831:
5827:
5822:
5820:
5818:
5814:
5811:
5810:Khattree 2022
5806:
5803:
5800:
5799:Khattree 2022
5795:
5792:
5789:
5788:Khattree 2019
5784:
5781:
5777:
5776:Latin squares
5771:
5768:
5763:
5759:
5754:
5749:
5745:
5741:
5740:
5735:
5728:
5725:
5721:
5716:
5713:
5709:
5704:
5701:
5695:
5691:
5688:
5686:
5683:
5682:
5678:
5676:
5673:
5662:
5658:
5647:
5644:
5641:
5638:
5637:
5633:
5630:
5627:
5624:
5623:
5619:
5616:
5613:
5610:
5609:
5605:
5602:
5599:
5596:
5595:
5592:
5590:
5580:
5573:
5566:
5562:
5561:
5560:
5553:
5549:
5542:
5536:
5520:
5513:
5502:
5499:
5496:
5495:
5491:
5488:
5485:
5484:
5480:
5477:
5474:
5473:
5469:
5466:
5463:
5462:
5458:
5455:
5452:
5451:
5447:
5444:
5441:
5440:
5436:
5433:
5430:
5429:
5426:
5424:
5420:
5416:
5411:
5404:
5402:
5400:
5396:
5392:
5388:
5383:
5381:
5375:
5373:
5369:
5361:
5356:
5353:
5350:
5347:
5344:
5341:
5340:
5339:
5337:
5334:PBIBD(2)s by
5333:
5325:
5323:
5296:
5291:
5288:
5284:
5278:
5274:
5270:
5265:
5260:
5257:
5253:
5247:
5243:
5235:
5219:
5215:
5211:
5206:
5201:
5198:
5194:
5188:
5183:
5180:
5177:
5173:
5165:
5148:
5145:
5142:
5136:
5133:
5128:
5124:
5118:
5114:
5108:
5103:
5100:
5097:
5093:
5085:
5071:
5068:
5065:
5062:
5057:
5053:
5047:
5042:
5039:
5036:
5032:
5024:
5010:
5007:
5004:
5001:
4998:
4991:
4990:
4989:
4987:
4979:
4977:
4975:
4971:
4951:
4945:
4940:
4935:
4930:
4925:
4920:
4913:
4908:
4903:
4898:
4893:
4888:
4881:
4876:
4871:
4866:
4861:
4856:
4849:
4844:
4839:
4834:
4829:
4824:
4817:
4812:
4807:
4802:
4797:
4792:
4785:
4780:
4775:
4770:
4765:
4760:
4754:
4744:
4726:
4720:
4715:
4710:
4705:
4700:
4695:
4690:
4685:
4678:
4673:
4668:
4663:
4658:
4653:
4648:
4643:
4636:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4510:
4505:
4500:
4495:
4490:
4485:
4480:
4475:
4469:
4459:
4454:
4447:
4440:
4433:
4417:
4413:
4409:
4405:
4395:
4392:
4389:
4386:
4385:
4381:
4378:
4375:
4372:
4371:
4368:
4366:
4356: 0
4354:
4349:
4344:
4341: 2
4339:
4336: 3
4334:
4329:
4327:
4324:
4323:
4317:
4314: 0
4312:
4307:
4304: 3
4302:
4299: 2
4297:
4294: 3
4292:
4290:
4287:
4286:
4280:
4275:
4272: 0
4270:
4267: 3
4265:
4262: 3
4260:
4257: 2
4255:
4253:
4250:
4249:
4245: 2
4243:
4240: 3
4238:
4235: 3
4233:
4230: 0
4228:
4223:
4218:
4216:
4213:
4212:
4208: 3
4206:
4203: 2
4201:
4198: 3
4196:
4191:
4186:
4181:
4179:
4176:
4175:
4171: 3
4169:
4166: 3
4164:
4159:
4154:
4149:
4146: 0
4144:
4142:
4139:
4138:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4115:
4112:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4074:
4072:
4070:
4065:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4011:
4007:
4002:
4000:
3996:
3992:
3974:
3969:
3966:
3962:
3958:
3953:
3948:
3945:
3941:
3932:
3924:
3920:
3916:
3912:
3908:
3905:depending on
3890:
3885:
3882:
3878:
3855:
3851:
3847:
3841:
3838:
3835:
3810:
3806:
3802:
3796:
3793:
3790:
3767:
3764:
3761:
3739:
3735:
3731:
3725:
3722:
3719:
3708:
3706:
3702:
3698:
3694:
3675:
3672:
3666:
3663:
3660:
3646:
3643:
3640:
3631:
3626:
3622:
3613:
3610:
3591:
3588:
3585:
3582:
3576:
3573:
3570:
3561:
3556:
3552:
3544:
3543:
3542:
3540:
3533:
3529:
3525:
3521:
3501:
3497:
3493:
3489:
3485:
3482:
3478:
3474:
3471:
3467:
3463:
3455:
3453:
3451:
3447:
3443:
3439:
3435:
3431:
3427:
3423:
3419:
3415:
3413:
3409:
3404:
3402:
3398:
3383:
3376:
3372:
3365:
3359:
3356:
3355:
3354:
3353:
3352:
3351:
3350:
3348:
3343:
3341:
3337:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3304:
3298:
3296:
3292:
3288:
3284:
3280:
3277:, i.e., a 3-(
3276:
3268:
3266:
3259:
3255:
3252:
3249:
3245:
3242:
3239:
3236:
3235:
3234:
3232:
3228:
3224:
3220:
3216:
3214:
3209:
3207:
3203:
3200: +
3199:
3195:
3190:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3156:
3154:
3151:
3147:
3140:
3136:
3132:
3128:
3125:is called an
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3091:
3087:
3084:
3080:
3076:
3073:) design and
3072:
3068:
3064:
3060:
3056:
3052:
3044:
3042:
3040:
3035:
3033:
3029:
3025:
3021:
3016:
3014:
3011:-design with
3010:
3005:
3003:
2999:
2996: ≤
2995:
2991:
2983:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2932:
2929:
2926:
2921:
2918:
2915:
2902:
2889:
2886:
2883:
2878:
2875:
2872:
2860:
2857:
2852:
2848:
2844:
2841:
2813:
2810:
2798:
2786:
2783:
2772:
2769:
2764:
2760:
2756:
2753:
2744:
2742:
2735:
2731:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2665:
2662:
2659:
2654:
2651:
2648:
2623:
2620:
2617:
2612:
2609:
2606:
2591:
2588:
2583:
2579:
2571:
2570:
2569:
2567:
2563:
2559:
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2483:
2479:
2471:
2467:
2465:
2463:
2459:
2458:affine planes
2454:
2451:
2449:
2446: +
2445:
2442: ≥
2441:
2437:
2433:
2429:
2424:
2422:
2418:
2414:
2406:
2404:
2402:
2397:
2383:
2380:
2377:
2357:
2354:
2351:
2348:
2328:
2325:
2322:
2319:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2282:
2280:
2276:
2272:
2268:
2265: ×
2264:
2259:
2255:
2251:
2247:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2208:
2206:
2204:
2196:
2193:
2190:
2187:
2186:Menon designs
2183:
2179:
2178:
2174:
2172:
2167:(2,11) – see
2166:
2162:
2159:
2155:
2154:
2150:
2146:
2142:
2141:Paley digraph
2138:
2137:Raymond Paley
2134:
2131:Paley biplane
2126:
2123:
2119:
2115:
2112:
2108:
2104:
2101:
2097:
2096:
2095:
2092:
2090:
2087: =
2086:
2082:
2078:
2074:
2070:
2067: =
2066:
2062:
2058:
2054:
2050:
2042:
2040:
2036:
2034:
2030:
2026:
2022:
2018:
2013:
2011:
2007:
2003:
1999:
1994:
1990:
1986:
1983: +
1982:
1978:
1974:
1970:
1965:
1963:
1959:
1956: +
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1902:
1896:
1893:
1890:
1884:
1881:
1878:
1875:
1872:
1865:
1864:
1863:
1862:
1861:
1859:
1855:
1851:
1846:
1838:
1836:
1833:
1831:
1827:
1806:
1800:
1797:
1794:
1788:
1785:
1779:
1776:
1773:
1767:
1760:
1759:
1758:
1757:
1756:
1753:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1702:
1700:
1692:
1675:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1631:
1626:
1621:
1616:
1611:
1606:
1601:
1594:
1589:
1584:
1579:
1574:
1569:
1564:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1483:
1478:
1473:
1468:
1463:
1458:
1453:
1446:
1441:
1436:
1431:
1426:
1421:
1416:
1409:
1401:
1400:
1399:
1397:
1396:corresponding
1393:
1385:
1384:
1383:
1376:
1375:
1374:
1355:
1349:
1344:
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1297:
1292:
1287:
1282:
1277:
1272:
1267:
1262:
1257:
1252:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1210:
1205:
1200:
1193:
1188:
1183:
1178:
1173:
1168:
1163:
1158:
1153:
1148:
1141:
1136:
1131:
1126:
1121:
1116:
1111:
1106:
1101:
1096:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1038:
1029:
1028:
1027:
1025:
1021:
1017:
1016:binary matrix
1014:
1010:
1006:
998:
997:
996:
994:
990:
986:
982:
978:
970:
968:
964:
962:
959: ≥
958:
954:
953:Ronald Fisher
950:
945:
943:
939:
936: +
935:
931:
927:
924: −
923:
919:
915:
912: −
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
862:
860:
856:
852:
848:
844:
839:
837:
833:
829:
825:
821:
817:
813:
809:
805:
801:
797:
793:
789:
786:the triples (
785:
766:
760:
757:
754:
748:
745:
739:
736:
733:
727:
720:
719:
718:
716:
712:
708:
704:
685:
682:
679:
676:
673:
670:
663:
662:
661:
659:
655:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
603:
591:
587:
585:
582:
581:
577:
575:
572:
571:
567:
565:
562:
561:
557:
555:
552:
551:
548:
544:
542:
539:
538:
535:
534:
533:
531:
527:
523:
519:
515:
511:
507:
503:
498:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
444:
440:
436:
432:
428:
424:
420:
416:
408:
406:
404:
400:
397:
394:
390:
386:
385:binary matrix
381:
367:
364:
361:
358:
355:
347:
343:
339:
335:
331:
327:
326:configuration
323:
319:
315:
311:
307:
299:
297:
295:
291:
287:
283:
279:
274:
272:
268:
264:
259:
257:
253:
249:
245:
241:
236:
234:
230:
225:
223:
219:
217:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
131:
123:
121:
119:
115:
111:
107:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
45:
44:combinatorial
38:
31:
27:
23:
19:
7109:Applications
6942:Block design
6941:
6860:
6811:
6791:
6768:, Springer,
6765:
6747:
6741:
6709:
6703:
6678:
6672:
6653:
6652:. Series A.
6647:
6641:
6621:
6598:
6576:
6555:
6537:
6504:
6500:
6496:
6476:
6456:
6430:
6424:
6404:
6382:
6364:
6360:
6339:
6333:
6309:
6283:
6264:
6263:. Series A.
6258:
6202:
6198:
6188:
6176:
6167:
6155:
6143:
6131:
6119:
6107:
6095:
6083:
6071:
6059:
6047:
6040:Stinson 2003
6035:
6023:
6011:
5999:
5992:Stinson 2003
5987:
5975:
5964:
5953:
5940:
5933:
5921:
5916:, pp.320-335
5909:
5882:
5875:Stinson 2003
5870:
5862:
5858:
5854:
5850:
5846:
5842:
5838:
5833:
5805:
5794:
5783:
5770:
5743:
5737:
5727:
5720:Stinson 2003
5715:
5703:
5674:
5663:
5656:
5653:
5586:
5578:
5571:
5564:
5551:
5547:
5540:
5537:
5518:
5511:
5508:
5418:
5412:
5408:
5384:
5376:
5365:
5362:Applications
5331:
5329:
5313:
4985:
4983:
4973:
4969:
4967:
4742:
4452:
4445:
4438:
4431:
4415:
4411:
4407:
4403:
4401:
4364:
4362:
4325:
4288:
4251:
4214:
4177:
4140:
4104:
4100:
4099:if elements
4096:
4092:
4088:
4084:
4080:
4078:
4068:
4066:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4021:
4020:-set X with
4017:
4013:
4009:
4005:
4003:
3998:
3994:
3990:
3930:
3928:
3922:
3918:
3914:
3910:
3906:
3704:
3700:
3696:
3692:
3538:
3531:
3527:
3523:
3519:
3495:
3491:
3487:
3483:
3476:
3472:
3461:
3459:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3417:
3416:
3407:
3405:
3400:
3394:
3381:
3374:
3370:
3363:
3357:
3344:
3339:
3335:
3331:
3327:
3323:
3319:
3315:
3311:
3307:
3301:
3299:
3294:
3291:Möbius plane
3286:
3282:
3278:
3275:affine plane
3272:
3263:
3257:
3253:
3247:
3243:
3237:
3230:
3226:
3222:
3218:
3217:
3212:
3210:
3205:
3201:
3197:
3191:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3157:
3152:
3149:
3145:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3089:
3085:
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3048:
3039:block design
3038:
3036:
3027:
3023:
3019:
3017:
3012:
3008:
3006:
3001:
2997:
2993:
2989:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2956:
2745:
2737:
2733:
2726:
2724:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2477:
2475:
2469:
2455:
2452:
2447:
2443:
2439:
2435:
2431:
2427:
2425:
2420:
2416:
2412:
2410:
2400:
2398:
2309:
2305:
2301:
2297:
2289:
2285:
2283:
2278:
2274:
2270:
2266:
2262:
2257:
2253:
2249:
2245:
2238:
2234:
2230:
2226:
2222:
2218:
2212:
2200:
2175:for details.
2170:
2164:
2160:
2128:
2121:
2110:
2106:
2093:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2046:
2037:
2032:
2028:
2024:
2016:
2014:
2009:
2005:
2001:
1997:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1966:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1921:
1919:
1857:
1853:
1848:
1834:
1825:
1823:
1754:
1749:
1745:
1741:
1737:
1733:
1729:
1721:
1717:
1713:
1709:
1705:
1703:
1698:
1696:
1389:
1381:
1372:
1023:
1019:
1012:
1008:
1002:
992:
988:
984:
980:
976:
974:
965:
960:
956:
946:
941:
937:
933:
929:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
881:
877:
873:
869:
865:
863:
858:
854:
850:
846:
842:
840:
835:
831:
827:
823:
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
781:
714:
710:
706:
702:
700:
657:
653:
649:
645:
641:
637:
633:
629:
625:
621:
617:
613:
609:
605:
601:
599:
589:
583:
573:
563:
553:
546:
540:
529:
525:
521:
517:
513:
509:
505:
501:
499:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
442:
438:
434:
430:
426:
422:
418:
414:
412:
382:
345:
341:
337:
333:
325:
313:
309:
305:
303:
293:
289:
281:
275:
262:
260:
251:
247:
243:
242:) is called
239:
237:
232:
228:
226:
215:
213:
209:
208:-value. For
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
127:
117:
109:
105:
101:
98:block design
97:
95:
87:cryptography
62:
51:block design
50:
41:
6980:Statistics
6330:Bose, R. C.
5947:, p. 4
5391:block codes
5345:triangular;
5322:is a BIBD.
4988:) satisfy:
4095:) entry is
3931:commutative
3077:a point of
2122:complements
47:mathematics
7136:Categories
7009:Fano plane
6974:Hypergraph
6448:2440/15239
6247:References
5887:Ryser 1963
5847:projective
5710:, pp.17−19
5597:Treatment
5437:Treatment
5419:design.bib
5417:-function
5332:then known
4980:Properties
3780:such that
3524:associates
3153:extendable
3148:; we call
3061:) be a t-(
2746:Note that
2546:parameters
2421:resolution
2118:Fano plane
2021:Fano plane
1392:Fano plane
955:, is that
882:complement
522:parameters
403:Levi graph
282:non-binary
278:statistics
229:incomplete
164:, denoted
26:statistics
6959:Incidence
6862:MathWorld
6726:225335042
6695:125795689
6579:. Dover.
6529:120721016
6433:: 52–75,
6212:1203.5378
6183:, pg. 127
6162:, pg. 242
6126:, pg. 238
6114:, pg. 237
5914:Hall 1986
5863:symmetric
5762:0195-6698
5174:∑
5146:−
5125:λ
5094:∑
5069:−
5033:∑
4367:(3) are:
3848:∈
3803:∈
3765:∈
3732:∈
3673:∈
3627:∗
3614:Defining
3589:∈
3522:th–
3481:partition
3127:extension
3037:The term
2930:−
2919:−
2887:−
2876:−
2861:λ
2849:λ
2773:λ
2761:λ
2701:…
2663:−
2652:−
2621:−
2610:−
2592:λ
2580:λ
2540:, λ, and
2496:, called
2472:-designs)
2381:−
2355:−
2326:−
2163:(2,5) in
1894:−
1876:−
1798:−
1777:−
1768:λ
758:−
737:−
728:λ
453:, called
396:bipartite
393:biregular
218:) designs
136:) if all
110:2-design,
61:known as
7073:Theorems
6984:Blocking
6954:Geometry
6789:(1987).
6575:(1988).
6308:(1986),
5904:, pg.109
5679:See also
5606:Block C
5603:Block B
5600:Block A
4976:values.
4067:A PBIBD(
4064:blocks.
3989:for all
3510:, ..., R
3475:of size
3306:in PG(3,
3161:: If a
2544:are the
2484:-design
2426:If a 2-(
2396:blocks.
2244:, where
2217:of size
2109:= 4 and
2043:Biplanes
971:Examples
798:) where
709:) where
435:2-design
322:geometry
314:1-design
271:multiset
258:(PBDs).
130:balanced
124:Overview
118:t-design
67:symmetry
6521:2384014
6499:= 11".
6237:7586742
6217:Bibcode
6054:, pg.29
5928:, pg.55
5828:, p. 27
5581:= {1, 2
5574:= {1, 3
5567:= {2, 3
5421:of the
5351:cyclic;
4075:Example
3699:, then
3537:
3490:×
3464:-class
3420:. Let
3418:Theorem
3340:egglike
3219:Theorem
3159:Theorem
2958:Theorem
2292:is the
2261:is the
2229:matrix
2049:biplane
2023:, with
880:. The
294:regular
244:uniform
196:if the
170:regular
132:(up to
6930:Fields
6818:
6799:
6785:&
6772:
6724:
6693:
6628:
6605:
6583:
6562:
6544:
6527:
6519:
6485:
6463:
6413:
6391:
6321:
6290:
6235:
5855:square
5760:
5576:} and
5434:Block
5431:Plots
4117:
3432:); so
3221::. If
3081:. The
2960:: Any
2725:where
2498:blocks
2221:is an
2173:points
2135:after
1920:Since
1026:) is:
857:, and
656:, and
455:blocks
419:points
383:Every
328:, see
263:simple
248:proper
214:PBIBD(
156:. For
89:, and
63:blocks
53:is an
28:, see
20:. For
6722:S2CID
6691:S2CID
6525:S2CID
6233:S2CID
6207:arXiv
5945:(PDF)
5722:, p.1
5696:Notes
5670:λ = 1
5525:λ = 1
4008:with
3691:, if
3494:into
3403:). .
3303:ovoid
3289:, or
2743:= λ.
2100:digon
1993:lines
1740:is a
1732:is a
1726:Ryser
1378:2456.
983:= 3,
979:= 6,
866:order
528:>
500:Here
399:graph
7064:Dual
6816:ISBN
6797:ISBN
6770:ISBN
6644:= 9"
6626:ISBN
6603:ISBN
6581:ISBN
6560:ISBN
6542:ISBN
6483:ISBN
6461:ISBN
6411:ISBN
6389:ISBN
6319:ISBN
6288:ISBN
5758:ISSN
5555:and
5531:and
5523:and
5497:302
5486:301
5475:202
5464:201
5453:102
5442:101
5385:The
4972:and
4103:and
4079:Let
4048:are
4044:and
3997:and
3921:and
3825:and
3530:has
3498:+ 1
3049:Let
2834:and
2564:and
2480:, a
2252:and
2015:For
1750:X, B
932:′ =
920:′ =
908:′ =
900:′ =
892:′ =
864:The
845:and
802:and
644:and
473:and
439:BIBD
437:(or
150:pair
106:BIBD
49:, a
6752:doi
6714:doi
6683:doi
6658:doi
6509:doi
6443:hdl
6435:doi
6369:doi
6344:doi
6269:doi
6225:doi
5841:or
5748:doi
5659:= 2
5543:= 3
5521:= 2
5514:= 3
3933:if
3709:If
3703:in
3695:in
3506:, R
3502:, R
3486:of
3470:set
3460:An
3406:If
3215:).
3133:if
3129:of
3053:= (
3004:.)
2528:),
2504:in
2213:An
2165:PSL
2161:PSL
2091:).
2051:or
1944:= (
1932:as
1007:(a
826:in
489:in
477:in
461:in
320:in
312:or
276:In
267:set
246:or
42:In
24:in
7138::
6859:.
6844::
6746:,
6736:;
6720:.
6710:51
6708:.
6689:.
6679:48
6677:.
6654:24
6646:.
6620:,
6523:.
6517:MR
6515:.
6505:16
6503:.
6441:,
6431:10
6429:,
6365:47
6363:,
6340:20
6338:,
6312:,
6304:;
6265:11
6257:.
6231:.
6223:.
6215:.
6203:16
6201:.
5894:^
5816:^
5756:.
5744:28
5742:.
5736:.
5672:.
5661:.
5648:0
5645:1
5642:1
5639:3
5634:1
5631:0
5628:1
5625:2
5620:1
5617:1
5614:0
5611:1
5583:}.
5550:,
5516:,
5503:1
5500:3
5492:2
5489:3
5481:3
5478:2
5470:1
5467:2
5459:2
5456:1
5448:3
5445:1
5401:.
5382:.
5338::
4135:6
4111:.
4056:≤
4004:A
3993:,
3913:,
3909:,
3701:R*
3632::=
3414:.
3387:),
3380:,
3369:+
3297:.
3165:-(
3057:,
3034:.
3022:-(
3018:A
2984:,λ
2976:-(
2964:-(
2954:.
2552:-(
2536:,
2411:A
2403:.
2235:HH
2225:×
2205:.
2047:A
2008:=
1979:=
1971:=
1964:.
1936:=
1924:=
1716:=
1708:=
928:,
916:,
904:,
896:,
872:=
853:,
794:,
790:,
705:,
652:,
636:,
628:,
624:,
620:,
616:,
608:,
604:,
516:,
429:,
425:,
405:.
273:.
224:.
120:.
93:.
85:,
81:,
77:,
73:,
6894:e
6887:t
6880:v
6865:.
6824:.
6805:.
6754::
6748:9
6728:.
6716::
6697:.
6685::
6666:.
6660::
6642:k
6611:.
6589:.
6531:.
6511::
6497:k
6445::
6437::
6371::
6346::
6325:.
6277:.
6271::
6239:.
6227::
6219::
6209::
5865:.
5764:.
5750::
5666:C
5657:r
5579:C
5572:B
5565:A
5557:C
5552:B
5548:A
5541:b
5533:r
5529:b
5519:k
5512:v
5415:R
5320:2
5316:1
5297:j
5292:h
5289:i
5285:p
5279:j
5275:n
5271:=
5266:i
5261:h
5258:j
5254:p
5248:i
5244:n
5220:j
5216:n
5212:=
5207:h
5202:u
5199:j
5195:p
5189:m
5184:0
5181:=
5178:u
5152:)
5149:1
5143:k
5140:(
5137:r
5134:=
5129:i
5119:i
5115:n
5109:m
5104:1
5101:=
5098:i
5072:1
5066:v
5063:=
5058:i
5054:n
5048:m
5043:1
5040:=
5037:i
5011:k
5008:b
5005:=
5002:r
4999:v
4986:m
4974:r
4970:λ
4952:)
4946:4
4941:2
4936:2
4931:2
4926:1
4921:1
4914:2
4909:4
4904:2
4899:1
4894:2
4889:1
4882:2
4877:2
4872:4
4867:1
4862:1
4857:2
4850:2
4845:1
4840:1
4835:4
4830:2
4825:2
4818:1
4813:2
4808:1
4803:2
4798:4
4793:2
4786:1
4781:1
4776:2
4771:2
4766:2
4761:4
4755:(
4727:)
4721:1
4716:1
4711:1
4706:0
4701:1
4696:0
4691:0
4686:0
4679:1
4674:1
4669:0
4664:1
4659:0
4654:0
4649:1
4644:0
4637:1
4632:1
4627:0
4622:0
4617:0
4612:1
4607:0
4602:1
4595:0
4590:0
4585:1
4580:1
4575:1
4570:1
4565:0
4560:0
4553:0
4548:0
4543:1
4538:1
4533:0
4528:0
4523:1
4518:1
4511:0
4506:0
4501:0
4496:0
4491:1
4486:1
4481:1
4476:1
4470:(
4456:3
4453:n
4449:1
4446:n
4442:2
4439:n
4435:0
4432:n
4428:3
4424:2
4420:1
4416:r
4412:k
4408:b
4404:v
4365:A
4326:6
4289:5
4252:4
4215:3
4178:2
4141:1
4132:5
4129:4
4126:3
4123:2
4120:1
4109:s
4105:j
4101:i
4097:s
4093:j
4091:,
4089:i
4085:X
4081:A
4069:n
4062:i
4058:n
4054:i
4050:i
4046:y
4042:x
4038:X
4034:n
4030:r
4026:k
4022:b
4018:v
4014:n
4010:n
3999:k
3995:j
3991:i
3975:k
3970:i
3967:j
3963:p
3959:=
3954:k
3949:j
3946:i
3942:p
3925:.
3923:y
3919:x
3915:k
3911:j
3907:i
3891:k
3886:j
3883:i
3879:p
3856:j
3852:R
3845:)
3842:y
3839:,
3836:z
3833:(
3811:i
3807:R
3800:)
3797:z
3794:,
3791:x
3788:(
3768:X
3762:z
3740:k
3736:R
3729:)
3726:y
3723:,
3720:x
3717:(
3705:S
3697:S
3693:R
3679:}
3676:R
3670:)
3667:x
3664:,
3661:y
3658:(
3654:|
3650:)
3647:y
3644:,
3641:x
3638:(
3635:{
3623:R
3611:.
3595:}
3592:X
3586:x
3583::
3580:)
3577:x
3574:,
3571:x
3568:(
3565:{
3562:=
3557:0
3553:R
3539:i
3535:i
3532:n
3528:X
3520:i
3516:i
3512:n
3508:1
3504:0
3496:n
3492:X
3488:X
3484:S
3477:v
3473:X
3462:n
3450:q
3446:q
3442:q
3438:q
3434:q
3430:q
3426:q
3422:q
3408:q
3401:q
3385:4
3382:x
3378:3
3375:x
3373:(
3371:f
3367:2
3364:x
3361:1
3358:x
3336:q
3332:O
3328:q
3324:q
3320:O
3316:q
3312:q
3308:q
3295:n
3283:n
3279:n
3258:k
3254:v
3248:k
3244:v
3238:D
3231:k
3229:,
3227:v
3223:D
3206:n
3202:n
3198:n
3187:v
3185:(
3183:b
3179:k
3175:λ
3173:,
3171:k
3169:,
3167:v
3163:t
3150:D
3146:D
3142:p
3139:E
3135:E
3131:D
3123:E
3119:λ
3115:k
3111:v
3107:t
3103:D
3099:p
3095:X
3090:p
3086:D
3079:X
3075:p
3071:λ
3069:,
3067:k
3065:,
3063:v
3059:B
3055:X
3051:D
3028:k
3026:,
3024:v
3020:t
3013:t
3009:t
3002:s
2998:t
2994:s
2990:s
2986:s
2982:k
2980:,
2978:v
2974:s
2970:k
2968:,
2966:v
2962:t
2939:)
2933:1
2927:t
2922:1
2916:k
2910:(
2903:/
2896:)
2890:1
2884:t
2879:1
2873:v
2867:(
2858:=
2853:1
2845:=
2842:r
2819:)
2814:t
2811:k
2806:(
2799:/
2792:)
2787:t
2784:v
2779:(
2770:=
2765:0
2757:=
2754:b
2740:t
2738:λ
2734:i
2729:i
2727:λ
2710:,
2707:t
2704:,
2698:,
2695:1
2692:,
2689:0
2686:=
2683:i
2672:)
2666:i
2660:t
2655:i
2649:k
2643:(
2636:/
2630:)
2624:i
2618:t
2613:i
2607:v
2601:(
2589:=
2584:i
2566:r
2562:b
2558:k
2556:,
2554:v
2550:t
2542:t
2538:r
2534:k
2530:b
2526:X
2522:v
2518:T
2514:t
2510:r
2506:X
2502:x
2494:X
2490:k
2486:B
2482:t
2478:t
2470:t
2448:c
2444:v
2440:b
2436:c
2432:k
2430:,
2428:v
2401:a
2384:1
2378:a
2358:1
2352:a
2349:2
2329:1
2323:a
2320:4
2306:a
2302:a
2298:a
2290:M
2286:a
2279:m
2275:m
2267:m
2263:m
2258:m
2254:I
2250:H
2246:H
2242:m
2239:I
2231:H
2227:m
2223:m
2219:m
2188:.
2171:p
2151:.
2111:k
2107:v
2102:.
2089:k
2085:r
2081:n
2077:n
2073:v
2069:n
2065:k
2061:n
2057:λ
2033:n
2029:n
2025:v
2017:n
2010:n
2006:r
2002:n
1998:k
1989:b
1985:n
1981:n
1977:b
1973:v
1969:b
1962:n
1958:n
1954:n
1950:n
1946:n
1942:v
1938:k
1934:n
1926:r
1922:k
1903:.
1900:)
1897:1
1891:k
1888:(
1885:k
1882:=
1879:1
1873:v
1858:n
1854:λ
1826:v
1807:.
1804:)
1801:1
1795:k
1792:(
1789:k
1786:=
1783:)
1780:1
1774:v
1771:(
1746:k
1742:v
1738:B
1734:v
1730:X
1722:λ
1718:v
1714:b
1710:k
1706:r
1676:)
1669:0
1664:1
1659:1
1654:0
1649:1
1644:0
1639:0
1632:1
1627:0
1622:0
1617:1
1612:1
1607:0
1602:0
1595:1
1590:0
1585:1
1580:0
1575:0
1570:1
1565:0
1558:0
1553:1
1548:0
1543:1
1538:0
1533:1
1528:0
1521:1
1516:1
1511:0
1506:0
1501:0
1496:0
1491:1
1484:0
1479:0
1474:1
1469:1
1464:0
1459:0
1454:1
1447:0
1442:0
1437:0
1432:0
1427:1
1422:1
1417:1
1410:(
1356:)
1350:1
1345:0
1340:1
1335:0
1330:1
1325:1
1320:1
1315:0
1310:0
1305:0
1298:0
1293:1
1288:1
1283:1
1278:0
1273:1
1268:0
1263:1
1258:0
1253:0
1246:1
1241:1
1236:0
1231:1
1226:0
1221:0
1216:1
1211:0
1206:1
1201:0
1194:1
1189:1
1184:0
1179:0
1174:1
1169:0
1164:0
1159:1
1154:0
1149:1
1142:0
1137:0
1132:1
1127:1
1122:1
1117:0
1112:0
1107:0
1102:1
1097:1
1090:0
1085:0
1080:0
1075:0
1070:0
1065:1
1060:1
1055:1
1050:1
1045:1
1039:(
1024:k
1020:r
1013:b
1011:×
1009:v
993:r
989:b
985:λ
981:k
977:v
961:v
957:b
942:r
938:b
934:λ
930:λ
926:k
922:v
918:k
914:r
910:b
906:r
902:b
898:b
894:v
890:v
886:X
878:λ
874:r
870:n
859:λ
855:k
851:v
847:r
843:b
836:r
832:r
828:X
824:x
820:x
816:r
812:x
808:B
804:y
800:x
796:B
792:y
788:x
784:x
767:,
764:)
761:1
755:k
752:(
749:r
746:=
743:)
740:1
734:v
731:(
715:p
711:B
707:p
703:B
686:,
683:r
680:v
677:=
674:k
671:b
658:λ
654:k
650:v
646:r
642:b
638:k
634:v
630:λ
626:k
622:r
618:b
614:v
610:λ
606:k
602:v
590:t
584:λ
574:k
564:r
554:b
547:X
541:v
530:k
526:v
518:r
514:k
510:b
506:X
502:v
495:r
491:X
487:x
483:λ
479:X
475:y
471:x
467:r
463:X
459:x
451:X
447:k
443:B
431:λ
427:r
423:k
415:X
368:r
365:v
362:=
359:k
356:b
346:b
342:v
338:r
334:k
306:t
252:t
240:k
233:k
216:n
210:t
206:λ
202:n
198:t
190:t
186:t
182:λ
178:t
174:t
166:r
158:t
146:t
142:λ
138:t
134:t
104:(
39:.
32:.
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